# Java data structure and algorithm analysis (nine) -- B tree

Original 2017, 09, 27, 09:41:56

# Introduction to B tree

## Definition

In computer science, the B tree (English: B-tree) is a self balancing tree that keeps data in order. This data structure enables data lookups, sequential access, insertion, and deletion operations to be done in log time.

## Characteristic

The B tree of order M is a tree with the following characteristics:
The 1. data item is stored on the leaves
2. non leaf nodes until the M-1 keyword indicates the direction of the search: the keyword I stands for the smallest keyword in the subtree i+1
The root of a 3. tree, or a leaf, or its son between 2 and M
4. except for all non leaf nodes the number of sons between M/2 and M.
5. all leaves are at the same depth and have data items between L/2 and L

For example: (M=3)

## Find and insert

For convenience, a special sentry key is used, which is smaller than all other keys and is represented by *.

In the beginning, the B tree contains only one root node, and the root node contains only the anchor key at initialization.

Each key in the inner node is associated with a node, and the subtree of this node is the root. All keys are greater than or equal to the key associated with this node, but less than all other keys.

# Implementation of B tree

PublicClass BTree2<K,V>
{
PrivateStatic, Log, logger = LogFactory.getLog (BTree.Class);

/ * *
* the key pair in the B tree node.
* <p/>
* the nodes in the B tree store key value pairs.
* access values by key.
*
*@param<K> - key type
*@param<V> - value type
* /
PrivateStaticClass Entry<K,V>
{
PrivateK key;
PrivateV value;

Public Entry (K, K, V, V)
{
This.key = k;
This.value = v;
}

Public, K, getKey ()
{
ReturnKey;
}

Public, V, getValue ()
{
ReturnValue;
}

Public void setValue (V value)
{
This.value = value;
}

@Override
Public, String, toString ()
{
ReturnKey +":"+ value;
}
}

/ * *
* searching for the return result of a given key value in the B tree node.
* <p/>
* the result consists of two parts. The first part indicates whether the search was successful,
* if the search succeeds, the second part represents the location of the given key value in the B tree node,
* if the lookup fails, the second part indicates where the given key value should be inserted.
* /
PrivateStaticClass SearchResult<V>
{
PrivateBoolean exist;
PrivateInt index;
PrivateV value;

Public SearchResult (Boolean, exist, int, index)
{
This.exist = exist;
This.index = index;
}

Public SearchResult (Boolean, exist, int, index, V, value)
{
This(exist, index);
This.value = value;
}

Public, Boolean, isExist ()
{
ReturnExist;
}

Public, int, getIndex ()
{
ReturnIndex;
}

Public, V, getValue ()
{
ReturnValue;
}
}

/ * *
* nodes in the B tree.
*
* TODO needs to consider access in concurrent cases.
* /
PrivateStaticClass BTreeNode<K,V>
{
A key node / * * * / store, non descending
PrivateList<Entry<K, V>>, entrys;
* / / * * node node
PrivateList<BTreeNode<K, V>>, children;
Whether the leaf node * / / * *
PrivateBoolean leaf;
The key comparison object * / / * *
PrivateComparator<K> kComparator;

PrivateBTreeNode ()
{
Entrys =NewArrayList<Entry<K, V>> ();
Children =NewArrayList<BTreeNode<K, V>> ();
= leafFalse;
}

Public BTreeNode (Comparator<K> kComparator)
{
This();
This.kComparator = kComparator;
}

Public, Boolean, isLeaf ()
{
ReturnLeaf;
}

Public void setLeaf (Boolean leaf)
{
This.leaf = leaf;
}

/ * *
* returns the number of entries. If it is a non leaf node, according to the definition of the B tree,
* the number of child nodes in this node is ({)@link#size ()} + 1).
*
*@returnNumber of keywords
* /
Public, int, size ()
{
ReturnEntrys.size ();
}

@SuppressWarnings("Unchecked."")
Int compare (K, key1, K, key2)
{
Return= = kComparatorNull((Comparable<K>) key1).CompareTo (key2): kComparator.compare (key1, key2);
}

/ * *
* find a given key in the node.
* if a given key exists in the node, a <code>SearchResult</code> is returned,
* identifies the lookup success, the index of the given key in the node, and the value associated with the given key;
* if it does not exist, return <code>SearchResult</code>,
* identifies this lookup failure where the given key should be inserted. The key associated value is null.
* <p/>
* if the lookup fails, the index field in the returned result is [0, {@link(#size)}];
* if the search succeeds, the index field in the returned result is [0, {@link#size ()} - 1]
* <p/>
* this is a two bit lookup algorithm that guarantees time complexity of O (log (T)).
*
*@paramKey - given key values
*@return- finding results
* /
Public SearchResult<V> searchKey (K key)
{
Int Low =Zero;
Int high = entrys.size () -One;
Int mid =Zero;
While(low high)
{
Mid = (low + high) /Two;/ / the first that writing and implementation of the BTree, l+h can not overflow
Entry<K, V>, entry = entrys.get (mid);
If(compare (entry.getKey), (key) = =Zero)/ / entrys.get (MID) (.GetKey = key)
Break;
Else If(compare (entry.getKey (), key) >Zero)/ / entrys.get (MID).GetKey (key).
High = mid -One;
Else / / entry.get (MID).GetKey (key).
Low = mid +One;
}
Boolean result =False;
Int index =Zero;
V value =Null;
If(low high)Show / find success
{
Result =True;
Index = mid;Index said the location where the elements / /
Value = entrys.get (index).GetValue ();
}
Else
{
Result =False;
Index = low;Index said the element should be inserted / / position
}
Return NewSearchResult<V> (result, index, value);
}

/ * *
* appends a given item to the end of the node,
* you need to ensure that after you call the method, the entry in the node is still
* stored in non descending order by keyword.
*
*@paramEntry - the given item
* /
Public, void, addEntry (Entry<K, V>, entry)
{
}

/ * *
* delete the <code>entry</code> of the given index.
* <p/>
* you need to ensure that the given index is legal.
*
*@paramIndex - given index
*@paramAn item at the given index
* /
Public, Entry<K, V>, removeEntry (int, index)
{
ReturnEntrys.remove (index);
}

/ * *
* gets the entry of the given index in the node.
* <p/>
* you need to ensure that the given index is legal.
*
*@paramIndex - given index
*@returnAn item of a given index in a node
* /
Public, Entry<K, V>, entryAt (int, index)
{
ReturnEntrys.get (index);
}

/ * *
* if the given key exists in the node, the value associated with it is updated.
* otherwise insert.
*
*@paramEntry - the given item
*@returnNull, if the key is not present before the node, otherwise the value associated with the return of the given key
* /
Public, V, putEntry (Entry<K, V>, entry)
{
SearchResult<V> result = searchKey (entry.getKey ());
If(result.isExist ())
{
V oldValue = entrys.get (result.getIndex ()).GetValue ();
Entrys.get (result.getIndex ()).SetValue (entry.getValue ());
ReturnOldValue;
}
Else
{
InsertEntry (entry, result.getIndex ());
Return Null;
}
}

/ * *
* inserts a given item in the node,
* this method ensures that the key value is stored in non descending order after insertion.
* <p/>
* however, the time complexity of the method is O (t).
* <p/>
* <b> note: key repetition is not allowed in the </b>B tree.
*
*@paramEntry - given key values
*@returnTrue, if the insert succeeds, false, if the insertion fails
* /
Public, Boolean, insertEntry (Entry<K, V>, entry)
{
SearchResult<V> result = searchKey (entry.getKey ());
If(result.isExist ())
Return False;
Else
{
InsertEntry (entry, result.getIndex ());
Return True;
}
}

/ * *
* inserts a given item at the location of the given index in the node,
* you need to make sure the entry is in the right place.
*
*@paramKey - given key values
*@paramIndex - given index
* /
Public, void, insertEntry (Entry<K, V>, entry, int, index)
{
*
* it's really disgusting to plug in by creating a new ArrayList, so let's go ahead
If only there was a reallocate like C.
* /
List<Entry<K, V>>, newEntrys =NewArrayList<Entry<K, V>> ();
Int i =Zero;
/ / index = 0 or index = (keys.size) have no problem
For(; I < index; + + I)
For(I; entrys.size); I < (+ +)
Entrys.clear ();
Entrys = newEntrys;
}

/ * *
* returns the child node of the given index in the node.
* <p/>
* you need to ensure that the given index is legal.
*
*@paramIndex - given index
*@returnThe child node corresponding to the given index
* /
Public, BTreeNode<K, V>, childAt (int, index)
{
If(isLeaf ())
Throw NewUnsupportedOperationException ("Leaf, node, doesn't, have, children.."");
ReturnChildren.get (index);
}

/ * *
* appends a given child node to the end of that node.
*
*@paramChild - given child nodes
* /
Public, void, addChild (BTreeNode<K, V>, child)
{
}

/ * *
* deletes a child node at the given index position in that node.
* </p>
* you need to ensure that the given index is legal.
*
*@paramIndex - given index
* /
Public void removeChild (int index)
{
Children.remove (index);
}

/ * *
* inserts a given child node into a given index in that node
* position.
*
*@paramChild - given child nodes
*@paramIndex - child node with insertion location
* /
Public, void, insertChild (BTreeNode<K, V>, child, int, index)
{
List<BTreeNode<K, V>>, newChildren =NewArrayList<BTreeNode<K, V>> ();
Int i =Zero;
For(; I < index; + + I)
For(I; children.size); I < (+ +)
Children = newChildren;
}
}

PrivateStaticFinalInt DEFAULT_T =Two;

The root node of the B tree / * * * /
PrivateBTreeNode<K, V>, root;
According to the definition of B / * * tree, the N keyword B tree number of each non root node satisfies (T - 1) < < = n (2t - 1).
PrivateInt t = DEFAULT_T;
The number of the smallest non key / * * * root node in the
PrivateInt minKeySize = t -One;
The number of the largest non key / * * * root node in the
PrivateInt maxKeySize =Two- *tOne;
The key comparison object * / / * *
PrivateComparator<K> kComparator;

/ * *
* construct a B tree, and use the natural ordering method for key values
* /
Public, BTree ()
{
= rootNewBTreeNode<K, V> ();
Root.setLeaf (True);
}

Public BTree (int t)
{
This();
This.t = t;
MinKeySize = t -One;
MaxKeySize =Two- *tOne;
}

/ * *
* constructing a B tree by comparing the function object with a given key value.
*
*@paramKComparator - key comparison function object
* /
Public BTree (Comparator<K> kComparator)
{
= rootNewBTreeNode<K, V> (kComparator);
Root.setLeaf (True);
This.kComparator = kComparator;
}

Public BTree (Comparator<K>, kComparator, int, t)
{
This(kComparator);
This.t = t;
MinKeySize = t -One;
MaxKeySize =Two- *tOne;
}

@SuppressWarnings("Unchecked."")
Int compare (K, key1, K, key2)
{
Return= = kComparatorNull((Comparable<K>) key1).CompareTo (key2): kComparator.compare (key1, key2);
}

/ * *
* searching for a given key.
*
*@paramKey - given key values
*@returnThe value associated with the key, if it exists, otherwise null
* /
Public V search (K key)
{
ReturnSearch (root, key);
}

/ * *
* recursive searches in subtrees with a given node as root
* given <code>key</code>
*
*@paramNode - the root node of the subtree
*@paramKey - given key values
*@returnThe value associated with the key, if it exists, otherwise null
* /
PrivateV search (BTreeNode<K, V>, node, K, key)
{
SearchResult<V> result = node.searchKey (key);
If(result.isExist ())
ReturnResult.getValue ();
Else
{
If(node.isLeaf ())
Return Null;
Else
Search (node.childAt (result.getIndex ()), key);

}
Return Null;
}

/ * *
* splitting a full child node <code>childNode</code>.
* <p/>
* you need to make sure that the given child node is full.
*
*@paramParentNode - parent node
*@paramChildNode - full child node
*@paramIndex - the index of the full child node in the parent node
* /
PrivateVoid, splitNode (BTreeNode<K, V>, parentNode, BTreeNode<K, V>, childNode, int, index)
{
Assert (childNode.size) = maxKeySize;

BTreeNode<K, V>, siblingNode =NewBTreeNode<K, V> (kComparator);
SiblingNode.setLeaf (childNode.isLeaf ());
The index will be / / child nodes for [t, 2T - 2] (T - 1) a new node is inserted
For(int, I =ZeroI; minKeySize; I) + +
/ / extraction middle term full subcategory node, which is index (T - 1)
Entry<K, V>, entry = childNode.entryAt (T -One);
Delete the index node in full / [t - 1, 2T - 2] t items
For(int, I = maxKeySize -OneI - > = t;One- - - I)
ChildNode.removeEntry (I);
If(... ChildNode.isLeaf ())/ / if full subcategory node is not a leaf node, it also need to deal with its child nodes
{
The index will be sub node / / [t, t sub node 2T - 1] insert the new node
For(int, I =ZeroI < minKeySize +OneI + +);
/ / delete index for full node [t, t sub node 2T - 1]
For(int i = maxKeySize; I > = t; I)
ChildNode.removeChild (I);
}
/ / entry will be inserted into the parent node
ParentNode.insertEntry (entry, index);
/ / insert the new node parent node
ParentNode.insertChild (siblingNode, index +)One);
}

/ * *
* inserts a given item in a non full node.
*
*@paramNode - non full node
*@paramEntry - the given item
*@returnTrue, if there is no item in the B tree, otherwise false
* /
PrivateBoolean, insertNotFull (BTreeNode<K, V>, node, Entry<K, V>, entry)
{
Assert node.size () < maxKeySize;

If(node.isLeaf ())/ / if is directly inserted into the leaf node,
ReturnNode.insertEntry (entry);
Else
{
Entry should be inserted in a given / * find the location of nodes, then the entry should be inserted
* in the subtree corresponding to this location
* /
SearchResult<V> result = node.searchKey (entry.getKey ());
If there is a direct return / /, failure
If(result.isExist ())
Return False;
BTreeNode<K, V>, childNode = node.childAt (result.getIndex ());
If(childNode.size) (= =Two- *tOne)If the node is full of node / /
{
The first division /
SplitNode (node, childNode, result.getIndex ());
If the new generation of the key / given entry key is greater than the split, we need to insert the new item on the right,
* otherwise left.
* /
If(compare (entry.getKey ()), node.entryAt (result.getIndex ()),.GetKey ())Zero)
ChildNode = node.childAt (result.getIndex () +)One);
}
ReturnInsertNotFull (childNode, entry);
}
}

/ * *
* insert a key key pair into the B tree.
*
*@paramKey - bond
*@paramValue - value
*@paramTrue, if there is no item in the B tree, otherwise false
* /
Public, Boolean, insert (K, key, V, value)
{
If(root.size) (= maxKeySize)/ / if the root node is full, then B trees grow taller
{
BTreeNode<K, V>, newRoot =NewBTreeNode<K, V> (kComparator);
NewRoot.setLeaf (False);
SplitNode (newRoot, root),Zero);
Root = newRoot;
}
ReturnInsertNotFull (root,NewEntry<K, V> (key, value);
}

/ * *
* if a given key is present, the value associated with the update key is updated,
* otherwise insert a given item.
*
*@paramNode - non full node
*@paramEntry - the given item
*@returnTrue, if there is no item in the B tree, otherwise false
* /
PrivateV, putNotFull (BTreeNode<K, V>, node, Entry<K, V>, entry)
{
Assert node.size () < maxKeySize;

If(node.isLeaf ())/ / if is directly inserted into the leaf node,
ReturnNode.putEntry (entry);
Else
{
Entry should be inserted in a given / * find the location of nodes, then the entry should be inserted
* in the subtree corresponding to this location
* /
SearchResult<V> result = node.searchKey (entry.getKey ());
If there is / / update.
If(result.isExist ())
ReturnNode.putEntry (entry);
BTreeNode<K, V>, childNode = node.childAt (result.getIndex ());
If(childNode.size) (= =Two- *tOne)If the node is full of node / /
{
The first division /
SplitNode (node, childNode, result.getIndex ());
If the new generation of the key / given entry key is greater than the split, we need to insert the new item on the right,
* otherwise left.
* /
If(compare (entry.getKey ()), node.entryAt (result.getIndex ()),.GetKey ())Zero)
ChildNode = node.childAt (result.getIndex () +)One);
}
ReturnPutNotFull (childNode, entry);
}
}

/ * *
* if the given key exists in the B tree, the update value is updated.
* otherwise insert.
*
*@paramKey - bond
*@paramValue - value
*@returnIf a given key exists in the B tree, the previous value is returned, otherwise null
* /
Public, V, put (K, key, V, value)
{
If(root.size) (= maxKeySize)/ / if the root node is full, then B trees grow taller
{
BTreeNode<K, V>, newRoot =NewBTreeNode<K, V> (kComparator);
NewRoot.setLeaf (False);
SplitNode (newRoot, root),Zero);
Root = newRoot;
}
ReturnPutNotFull (root,NewEntry<K, V> (key, value);
}

/ * *
* removes an item associated with a given key from the B tree.
*
*@paramKey - given key
*@returnIf an item associated with a given key exists in the B tree, the deleted item is returned, otherwise null
* /
Public, Entry<K, V>, delete (K, key)
{
ReturnDelete (root, key);
}

/ * *
* deletes an item associated with a given key from the subtree with a given <code>node</code> as the root.
* <p/>
* delete the implementation of the idea, please refer to the introduction to algorithms, the second edition of the eighteenth chapter.
*
*@paramNode - given nodes
*@paramKey - given key
*@returnIf an item associated with a given key exists in the B tree, the deleted item is returned, otherwise null
* /
PrivateEntry<K, V>, delete (BTreeNode<K, V>, node, K, key)
{
The process / need to ensure that the implementation of the non root node deletion operation, the key number is at least t.
Assert node.size (T) node || > = = = root;

SearchResult<V> result = node.searchKey (key);
*
* because this is the case of successful search, 0 (result.getIndex) (node.size < < = (- 1)),
* therefore, (result.getIndex () + 1) will not overflow.
* /
If(result.isExist ())
{
If the key nodes in node / / 1., and is a leaf node, then delete.
If(node.isLeaf ())
ReturnNode.removeEntry (result.getIndex ());
Else
{
If the node in node / / 2.a to key before the child node contains at least t items
BTreeNode<K, V>, leftChildNode = node.childAt (result.getIndex ());
If(leftChildNode.size) (> = t)
{
The use of leftChildNode / / the last item instead need to delete the items in the node
Node.removeEntry (result.getIndex ());
Node.insertEntry (leftChildNode.entryAt (leftChildNode.size ()) -One) (result.getIndex ());
/ / a recursive delete left child node in the last item
ReturnDelete (leftChildNode, leftChildNode.entryAt (leftChildNode.size ()) -One(.GetKey ());
}
Else
{
If the node in node / / 2.b at key after the child node contains at least t keyword
BTreeNode<K, V>, rightChildNode = node.childAt (result.getIndex () +)One);
If(rightChildNode.size) (> = t)
{
The use of rightChildNode / / the first item to delete instead of the items in the node
Node.removeEntry (result.getIndex ());
Node.insertEntry (rightChildNode.entryAt ()Zero) (result.getIndex ());
Right recursive delete the first item / sub node in the
ReturnDelete (rightChildNode, rightChildNode.entryAt)Zero(.GetKey ());
}
Else Before and after key / / 2.c to key node contains only a T-1
{
Entry<K, V>, deletedEntry = node.removeEntry (result.getIndex ());
Node.removeChild (result.getIndex () +One);
/ / will merge associated with key in node and rightChildNode in leftChildNode
For(int, I =ZeroI; rightChildNode.size; I) < (+ +)
/ / will merge into leftChildNode sub node in rightChildNode, if any
If(... RightChildNode.isLeaf ())
{
For(int, I =ZeroI; rightChildNode.size; I) (< = + +)
}
ReturnDelete (leftChildNode, key);
}
}
}
}
Else
{
*
* because it is the failure to find, 0 < = result.getIndex (< = node.size) (),
* therefore (result.getIndex () + 1) will overflow.
* /
If(node.isLeaf ())/ / if key is not in the node node, and is a leaf node, not what to do, because the key is not in the B tree
{
Logger.info ("The key:.""+ key +"Isn't, in, this, BTree.."");
Return Null;
}
BTreeNode<K, V>, childNode = node.childAt (result.getIndex ());
If(childNode.size) (> = t)If a node / / / / no less than t, then recursively delete
ReturnDelete (childNode, key);
Else / / 3
{
To find the right node / brother
BTreeNode<K, V>, siblingNode =Null;
Int, siblingIndex = -One;
If(result.getIndex () < node.size ())Right / / brother node
{
If(node.childAt (result.getIndex () +)One) (.Size) > = t)
{
SiblingNode = node.childAt (result.getIndex () +)One);
SiblingIndex = result.getIndex () +One;
}
}
/ / if brother node right does not meet the conditions, then try to the left of the brother node
If(siblingNode = =Null)
{
If(result.getIndex () >Zero)Left / / brother node
{
If(node.childAt (result.getIndex ()) -One) (.Size) > = t)
{
SiblingNode = node.childAt (result.getIndex ()) -One);
SiblingIndex = result.getIndex () -One;
}
}
}
3.a / / there is an adjacent sibling node contains at least t items
If(siblingNode = =!Null)
{
If(siblingIndex < result.getIndex ())/ / left brother node to meet the conditions
{
ChildNode.insertEntry (node.entryAt (siblingIndex)),Zero);
Node.removeEntry (siblingIndex);
Node.insertEntry (siblingNode.entryAt (siblingNode.size ()) -One) (siblingIndex);
SiblingNode.removeEntry (siblingNode.size () -One);
/ / the last child left siblings moved to childNode
If(... SiblingNode.isLeaf ())
{
ChildNode.insertChild (siblingNode.childAt (siblingNode.size ()),Zero);
SiblingNode.removeChild (siblingNode.size ());
}
}
Else Right / brother node to meet the conditions
{
ChildNode.insertEntry (node.entryAt (result.getIndex ()), childNode.size () -One);
Node.removeEntry (result.getIndex ());
Node.insertEntry (siblingNode.entryAt ()Zero) (result.getIndex ());
SiblingNode.removeEntry (Zero);
The first child / right sibling node to move to childNode
/ / childNode.insertChild (siblingNode.childAt) (0), childNode.size (+ 1);
If(... SiblingNode.isLeaf ())
{
SiblingNode.removeChild (Zero);
}
}
ReturnDelete (childNode, key);
}
Else / / 3.b if its adjacent node contains a T-1.
{
If(result.getIndex () < node.size ())Right there / brother, additional directly behind
{
BTreeNode<K, V>, rightSiblingNode = node.childAt (result.getIndex () +)One);
Node.removeEntry (result.getIndex ());
Node.removeChild (result.getIndex () +One);
For(int, I =ZeroI; rightSiblingNode.size; I) < (+ +)
If(... RightSiblingNode.isLeaf ())
{
For(int, I =ZeroI; rightSiblingNode.size; I) (< = + +)
}
}
Else / / left in front of the insertion node
{
BTreeNode<K, V>, leftSiblingNode = node.childAt (result.getIndex ()) -One);
ChildNode.insertEntry (node.entryAt (result.getIndex ()) -One),Zero);
Node.removeEntry (result.getIndex () -One);
Node.removeChild (result.getIndex () -One);
For(int, I = leftSiblingNode.size () -One> = I;Zero- - - I)
ChildNode.insertEntry (leftSiblingNode.entryAt (I)),Zero);
If(... LeftSiblingNode.isLeaf ())
{
For(int) (I = leftSiblingNode.size; I =Zero- - - I)
ChildNode.insertChild (leftSiblingNode.childAt (I)),Zero);
}
}
/ / if node is root and node do not contain any items.
If(node = = root & & node.size (= =)Zero)
Root = childNode;
ReturnDelete (childNode, key);
}
}
}
}

/ * *
* a simple hierarchical traversal of the B tree is implemented for outputting the B tree.
* /
Public, void, output ()
{
Queue<BTreeNode<K, V>>, queue =NewLinkedList<BTreeNode<K, V>> ();
Queue.offer (root);
While(... Queue.isEmpty ())
{
BTreeNode<K, V>, node = queue.poll ();
For(int, I =ZeroI; node.size; I) < (+ +)
System.out.print (node.entryAt (I) +"");
System.out.println ();
If(... Node.isLeaf ())
{
For(int, I =ZeroI; node.size; I) (< = + +)
Queue.offer (node.childAt (I));
}
}
}

Public, static, void, main (String[], args)
{
Random random =NewRandom ();
BTree2<Integer, Integer>, BTREE =NewBTree2<Integer, Integer>Three);
List<Integer> save =NewArrayList<Integer> ();
For(int, I =ZeroI; "TenI + +);
{
Int r = random.nextIntOne hundred);
System.out.println (R);
Btree.insert (R, R);
}

System.out.println ("Conducting");
Btree.output ();
System.out.println ("Conducting");
Btree.delete (save.get ()Zero));
Btree.output ();
}
}

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